I have 6 siblings as well and dh has 3... I would be hurt if they didn't call to tell my kids Happy Birthday!!! I call all 11 of my nieces and nephews on their birthdays and send them gifts even with no party. That's the joy of being an Auntie and having an Auntie!!! My mom was one of 7 and my dad one of 4 and the married ones always sent gifts for bdays and Christmas or at least took us out to do some activity (which was even better since it meant individual attention). It's the same way for us now.

I have 4 kids and am busy too, but I, personally, feel it's part of being a family (taking time out to relate and talk to one another). One of my sisters conjoins bday gifts with Christmas gifts and I'm fine with that, but she still calls on the day of to sing happy birthday to the kids. We just write it on our calendars to remember. We don't spend much on each other's kids' gifts... maybe $5-10 dollars at most, but it's the thought and even kids love a dollar store coloring book and crayons or a gift card for a free Kid's Meal or a ticket to a movie theater. A lot of times the birthday gifts are late in the mail, but that just stretches out the fun and surprise.
We do a combined birthday party for all my kids because 4 parties a year is just too expensive. It doesn't mean the day of is not important by any means.

To the OP; I would ask when your kiddo's birthday is coming up, "Hey, Sis, can you call to sing her 'happy birthday' on her big day? It'd mean the world to her." It takes 4 minutes out of one's day (and I can totally see how an older kid would be hurt if no one calls them on their big day). As far as gifts, I wouldn't bring it up.

I do have one sister-in-law who has not even acknowledged 2 out of my 4 kids' existence, doesn't do gifts for any of them, doesn't even really talk to me (she has issues, but that is a whole other can of worms... I wish her jealousy wouldn't get in the way of how she doesn't relate to my kids ). Anyway, I still send her children gifts and talk to them. It's not their fault their mother is psycho and hurtful to my children (yes, I think it's hurtful behavior with the ignoring). Sigh...

I have 18 nieces and nephews and several of them are my age and ttc. There is simply no way myself or my siblings can keep up with that many birthdays nor afford that many gifts, so we mutually decided many years ago birthdays aren't a big deal. We are all still very close and love each other no less. The same was true for my mothers family (she is one of 15 children), and they too are all very close. Buying each other stuff isn't what makes a family. It's a nice thought if that's what you want to do, I just don't think it is something worth getting upset over.